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arXiv · 2406.08890

On the number of zeros of $\mathop{\mathcal R}(s)$

Juan Arias de Reyna
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Abstract

We prove that the number of zeros $\varrho=\beta+i\gamma$ of $\mathop{\mathcal R}(s)$ with $0<\gamma\le T$ is given by \[N(T)=\frac{T}{4\pi}\log\frac{T}{2\pi}-\frac{T}{4\pi}-\frac12\sqrt{\frac{T}{2\pi}}+O(T^{2/5}\log^2 T).\] Here $\mathop{\mathcal R}(s)$ is the function that Siegel found in Riemann's papers. Siegel related the zeros of $\mathop{\mathcal R}(s)$ to the zeros of Riemann's zeta function. Our result on $N(T)$ improves the result of Siegel.

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Juan Arias de Reyna. 2024-06-13. On the number of zeros of $\mathop{\mathcal R}(s)$. https://arxiv.org/abs/2406.08890

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